# A rectangle cut into two pieces, which build a square

A rectangle with side length a and b are in ratio $$a : b = (n+1)^2 : n^2$$, where n is a positive integer.

Is it possible to cut each such rectangle into two pieces, which can be put together to build a square?

Yes, this is always possible

Proof:

This solution, shown with $$n = 6$$, can be applied for all $$n$$. The rectangle with dimensions $$n^2$$ by $$(n+1)^2$$ is divided into $$n\times(n+1)$$ congruent rectangles with dimensions $$n$$ by $$n+1$$ such that there are $$n$$ rows of height $$n$$ and $$n+1$$ columns of width $$n+1$$. The rectangle is cut into two congruent pieces along a stair-step diagonal. These pieces can be arranged to form a square with edge length $$n\times(n+1)$$, divided into $$n+1$$ rows of height $$n$$ and $$n$$ columns of width $$n+1$$.

• The cut is very hard to make out in your picture. Could you color it stronger? And maybe add a text description of what you are doing for visually impaired users ? Dec 3, 2021 at 10:32
• Your rectangle seems to have side in ratio n:(n=1) - 6 and 7 specifically. The case for n=6 would have sides of lengths 49 and 36, which is not the same shape, and does not enable a simplistic 1x1 "staircase" cut. Dec 3, 2021 at 12:44
• @AdamV if you read the (less than 20 words of) explanation given, you'll notice that the non-square looking small rectangles (that go into the square 6 times horizontally and 7 times vertically) do not actually represent unit squares.
– Bass
Dec 3, 2021 at 14:44
• @Bass it's not less than 20 words ;-P Dec 3, 2021 at 19:57
• @Falco Your suggestions have been applied. Dec 3, 2021 at 22:48

I managed to do it like this:

Start with a rectangle of size 16 x 9 and cut into two parts as shown below:

Then place the pieces together like this to form a 12 x 12 square:

• The question is asking if this is possible for all n. Dec 2, 2021 at 23:51
• @DanielMathias OK I see the "each" in the question now. Looks like you supplied the correct answer for that. Now I wonder if other rectangles can be cut/pasted into squares or only these kinds?
– JS1
Dec 2, 2021 at 23:57
• The steps of this staircase are each n x (n+1). That will always work because there will always be n divisions of n into n^2, and n+1 divisions of n+1 into (n+1)^2, enabling the "shift and lift" to equalise them so you end up with sides of (n+1) x n and n x (n=1) which are of course the same. Dec 3, 2021 at 12:48